The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli [quantph/0610153], we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko [Phys. Rev. A 88 012311 (2013)], we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis [New J. Phys. 12 083039 (2010)]. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.

翻译：CSS码构造是一个强大的框架，用于通过一对底层经典码来表达量子码的特性。其子系统扩展允许类似的表达，但一般情况尚未得到充分探索。在Aly、Klappenecker和Sarvepalli先前工作[quant-ph/0610153]的基础上，我们确定了子系统CSS码的参数，表达了码字，并仅利用两个底层经典码的数据开发了Steane型解码器。推广Kovalev和Pryadko的结果[Phys. Rev. A 88 012311 (2013)]，我们证明任何子系统稳定子码都可以通过“加倍”构造出一个子系统CSS码，该码具有两倍的物理量子比特、逻辑量子比特和规范量子比特数量，且码距最多可达两倍。该映射保持局域性，且比Bravyi、Terhal和Leemhuis基于马约拉纳的映射[New J. Phys. 12 083039 (2010)]更紧凑。利用古尔萨引理，我们证明每个子系统稳定子码都可以从两个满足特定约束的嵌套子系统CSS码构造出来，并根据嵌套码的性质刻画了子系统稳定子码的特征。